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G = C84D4order 64 = 26

1st semidirect product of C8 and D4 acting via D4/C4=C2

p-group, metabelian, nilpotent (class 3), monomial

Aliases: C84D4, C41D8, C42.80C22, (C4×C8)⋊8C2, (C2×D8)⋊5C2, C4.2(C2×D4), C41D43C2, C2.10(C2×D8), (C2×C4).77D4, C2.6(C41D4), (C2×C8).78C22, (C2×C4).118C23, (C2×D4).29C22, C22.114(C2×D4), SmallGroup(64,174)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — C84D4
C1C2C22C2×C4C42C4×C8 — C84D4
C1C2C2×C4 — C84D4
C1C22C42 — C84D4
C1C2C2C2×C4 — C84D4

Generators and relations for C84D4
 G = < a,b,c | a8=b4=c2=1, ab=ba, cac=a-1, cbc=b-1 >

Subgroups: 193 in 81 conjugacy classes, 33 normal (7 characteristic)
C1, C2, C2 [×2], C2 [×4], C4 [×6], C22, C22 [×12], C8 [×4], C2×C4, C2×C4 [×2], D4 [×16], C23 [×4], C42, C2×C8 [×2], D8 [×8], C2×D4 [×4], C2×D4 [×4], C4×C8, C41D4 [×2], C2×D8 [×4], C84D4
Quotients: C1, C2 [×7], C22 [×7], D4 [×6], C23, D8 [×4], C2×D4 [×3], C41D4, C2×D8 [×2], C84D4

Character table of C84D4

 class 12A2B2C2D2E2F2G4A4B4C4D4E4F8A8B8C8D8E8F8G8H
 size 1111888822222222222222
ρ11111111111111111111111    trivial
ρ2111111-1-1-1-11-11-1-11-11-11-11    linear of order 2
ρ311111-1-11111111-1-1-1-1-1-1-1-1    linear of order 2
ρ411111-11-1-1-11-11-11-11-11-11-1    linear of order 2
ρ51111-11-11-1-11-11-11-11-11-11-1    linear of order 2
ρ61111-111-1111111-1-1-1-1-1-1-1-1    linear of order 2
ρ71111-1-111-1-11-11-1-11-11-11-11    linear of order 2
ρ81111-1-1-1-111111111111111    linear of order 2
ρ92-2-22000000-202002020-20-2    orthogonal lifted from D4
ρ102-2-22000000-20200-20-20202    orthogonal lifted from D4
ρ112-2-2200000020-202020-20-20    orthogonal lifted from D4
ρ122-2-2200000020-20-20-202020    orthogonal lifted from D4
ρ1322220000-22-22-2-200000000    orthogonal lifted from D4
ρ14222200002-2-2-2-2200000000    orthogonal lifted from D4
ρ152-22-200000-20200-222-2-222-2    orthogonal lifted from D8
ρ1622-2-2000020000-2-222-22-2-22    orthogonal lifted from D8
ρ1722-2-2000020000-22-2-22-222-2    orthogonal lifted from D8
ρ182-22-200000-202002-2-222-2-22    orthogonal lifted from D8
ρ192-22-20000020-20022-2-222-2-2    orthogonal lifted from D8
ρ2022-2-20000-20000222-2-2-2-222    orthogonal lifted from D8
ρ212-22-20000020-200-2-222-2-222    orthogonal lifted from D8
ρ2222-2-20000-200002-2-22222-2-2    orthogonal lifted from D8

Smallest permutation representation of C84D4
On 32 points
Generators in S32
(1 2 3 4 5 6 7 8)(9 10 11 12 13 14 15 16)(17 18 19 20 21 22 23 24)(25 26 27 28 29 30 31 32)
(1 29 11 18)(2 30 12 19)(3 31 13 20)(4 32 14 21)(5 25 15 22)(6 26 16 23)(7 27 9 24)(8 28 10 17)
(1 11)(2 10)(3 9)(4 16)(5 15)(6 14)(7 13)(8 12)(17 19)(20 24)(21 23)(26 32)(27 31)(28 30)

G:=sub<Sym(32)| (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16)(17,18,19,20,21,22,23,24)(25,26,27,28,29,30,31,32), (1,29,11,18)(2,30,12,19)(3,31,13,20)(4,32,14,21)(5,25,15,22)(6,26,16,23)(7,27,9,24)(8,28,10,17), (1,11)(2,10)(3,9)(4,16)(5,15)(6,14)(7,13)(8,12)(17,19)(20,24)(21,23)(26,32)(27,31)(28,30)>;

G:=Group( (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16)(17,18,19,20,21,22,23,24)(25,26,27,28,29,30,31,32), (1,29,11,18)(2,30,12,19)(3,31,13,20)(4,32,14,21)(5,25,15,22)(6,26,16,23)(7,27,9,24)(8,28,10,17), (1,11)(2,10)(3,9)(4,16)(5,15)(6,14)(7,13)(8,12)(17,19)(20,24)(21,23)(26,32)(27,31)(28,30) );

G=PermutationGroup([(1,2,3,4,5,6,7,8),(9,10,11,12,13,14,15,16),(17,18,19,20,21,22,23,24),(25,26,27,28,29,30,31,32)], [(1,29,11,18),(2,30,12,19),(3,31,13,20),(4,32,14,21),(5,25,15,22),(6,26,16,23),(7,27,9,24),(8,28,10,17)], [(1,11),(2,10),(3,9),(4,16),(5,15),(6,14),(7,13),(8,12),(17,19),(20,24),(21,23),(26,32),(27,31),(28,30)])

C84D4 is a maximal subgroup of
C8.24D8  C4.D16  C41D4⋊C4  D4⋊D8  C42.181C23  Q8⋊D8  C813SD16  C82SD16  C825C2  D82D4  Q162D4  D83D4  C4.4D16  C8.13SD16  C42.263D4  D8○D8  Q85D8  C42.530C23
 C8p⋊D4: C4⋊D16  C165D4  C163D4  C124D8  C245D4  C204D8  C405D4  C284D8 ...
 C4p⋊D8: C87D8  C82D8  C85D8  C84D8  C83D8  C12⋊D8  C20⋊D8  C28⋊D8 ...
 C8pD4⋊C2: C86SD16  C42.664C23  C42.360D4  M4(2)⋊7D4  M4(2)⋊11D4  C42.366D4  C42.388C23  C42.261D4 ...
C84D4 is a maximal quotient of
C825C2  C84Q16  C8.2D8  C42.59Q8  C42.432D4  (C2×C4)⋊6D8  (C2×C4)⋊2D8  (C2×C4).27D8  C4⋊Q32  C8.21D8  C8.7D8
 C8p⋊D4: C4⋊D16  C165D4  C163D4  C124D8  C245D4  C204D8  C405D4  C284D8 ...
 C4p⋊D8: C85D8  C84D8  C83D8  C12⋊D8  C20⋊D8  C28⋊D8 ...

Matrix representation of C84D4 in GL4(𝔽17) generated by

31400
3300
00314
0033
,
16000
01600
0001
00160
,
1000
01600
00160
0001
G:=sub<GL(4,GF(17))| [3,3,0,0,14,3,0,0,0,0,3,3,0,0,14,3],[16,0,0,0,0,16,0,0,0,0,0,16,0,0,1,0],[1,0,0,0,0,16,0,0,0,0,16,0,0,0,0,1] >;

C84D4 in GAP, Magma, Sage, TeX

C_8\rtimes_4D_4
% in TeX

G:=Group("C8:4D4");
// GroupNames label

G:=SmallGroup(64,174);
// by ID

G=gap.SmallGroup(64,174);
# by ID

G:=PCGroup([6,-2,2,2,-2,2,-2,121,247,362,86,963,117]);
// Polycyclic

G:=Group<a,b,c|a^8=b^4=c^2=1,a*b=b*a,c*a*c=a^-1,c*b*c=b^-1>;
// generators/relations

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Character table of C84D4 in TeX

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